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4-Manifolds with Indefinite Intersection Form

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4-Manifolds with Indefinite Intersection Form 4-MANIFOLDS WITH INDEFINITE INTERSECTION FORM S.K. Donaldson All Souls College Oxford, England In writing up this lecture I shall not concentrate so much on des- cribing problems of 4-manifold topology; instead I shall explain how a simple topological construction has applications in two different direc- tions. First I will recall that, just as bundles over a single space have homotopy invariants, so do families of bundles, and that these define cor- responding invariants in families of connections. Next I will sketch the way in which such a topological invariant, when endowed with a geometric realisation, becomes important for studying holomorphic bundles over al- gebraic varieties. Last I will indicate how this same homotopy invariant of families of connections, combined with arguments involving moduli spa- ces of self-dual connections over a Riemannian 4-manifold, gives restric- tions on the possible homotopy types of smooth 4-manifolds and I will speculate on possible future progress in this area. Topology of bundles. This is standard material that may be found in [2] for example. Con- sider a fixed manifold X and a family of bundles over X parametrised by some auxiliary space T , so we have a bundle P over the product X × T with structure group G (compact and connected, say). Take first the case when T is a point so we have a single bundle over X , deter- mined up to equivalence by a homotopy class of maps from X to BG . This may be non-trivial, detected for example by characteristic classes in the cohomology of X . If we choose a connection A on the bundle the real characteristic classes can be represented by explicit differen- tial forms built from the curvature of the connection. Equally if D is an elliptic differential operator over X then using a connection it may be extended to act on objects (functions, forms, spinors etc.) twisted by a vector bundle associated to P . This has an integer valued index: index (DA) = dim ker D A - dim coker D A 310 which is a rigid invariant of the bundle, independent of the connection. So these are two ways in which the underlying homotopy may be represen- ted geometrically, by curvature and by differential operators. The Chern- Weil and Atiyah-Singer theorems then give formulae relating the three. In the same way for a general family parametrised by T the bundle P is classified by a homotopy class of maps from T to the mapping space Maps(X,BG) , and at the other extreme from the case T = point we have a universal family parametrised by this mapping space. Again we may al- ways choose a connection over X × T , which we may think of as a family of connections parametrised by T, and conversely any family of equiva- lence classes of connections on some bundle essentially arises in this way. (This is precisely true if we work with based maps and bundles, re- moving base points gives small technical differences which can safely be ignored here). Equivalently we have the infinite dimensional space B of all equivalence classes of connections obtained by dividing the affine space of connections A by the bundle automorphism group G . B has the homotopy type of Maps(X,BG) . Again we may construct topological invariants of such families of bundles. In cohomology we can use the characteristic classes again. There is a slant product: H p+q(x × T) ® H (X) --> H p(T) q so that characteristic classes of bundles over X x T contracted with, or integrated over, homology classes in the base manifold X yield coho- mology classes in families of connections. In particular if G is, say, a unitary group we obtain in this way a map: : H 2 (X) > H 2 (T) ~(~) = c2 (P)/~ (A simpler example is to take the Jacobian parametrising complex line bund- les over a Riem~nn surface. Operating in the same way with the first Chern class gives the usual correspondence between the 1-dimensional homology of the surface and the cohomology of the Jacobian). We can do the corres- 311 ponding thing in K-theory and realise the resulting elements in the K- theory of T by using differential operators again. For example if the base manifold X is the 2-sphere then a unitary bundle over S 2 x T de- fines an element of K(S 2 × T) which maps to K(T) by the inverse of the Bott periodicity map. If we take the Dirac operator D over S 2 then a family of connections gives a family of Dirac operators {D t} parame- trised by T and, after suitable stabilisation the index of this family [2] defines the required class: index D t = [Ker D t] - [coker D t] 6 K(T) Of course we obtain other classes in this way and the Atiyah-Singer index theorem for families gives formulae relating these to the underlying ho- motopy. In particular we may understand our class above from either point of view via the formula: c1(index D t) : ~(fundamental class of S 2) S_table bundles on algebraic curves and surfaces ~ Here I only want to say enough to fit into our overall theme; more details and references may be found in [4], but I learnt the point of view we are adopting now from lectures of Quillen. There is a general algebraic theory dealing with the action of a complex reductive group G ~ on a vector space ~n+1 via a linear repre- sentation. Equivalently we may take the induced action on ~n and the hyperplane bundle H over it. In that theory there is a definition of a "stable" point. Now suppose that ~n+1 has a fixed Hermitian metric, in- ducing metrics on H and on ~pn , and picking out a maximal compact subgroup GcG ~ whose action preserves these metrics. There is a general theory dealing with the metrical properties of these actions and relating them to the purely complex algebraic properties. Roughly speaking if we restrict to the stable points then a transversal to the G~-action on ~n is induced by taking the points in ~n+1 , or equivalently H -I , which minimise the norm in their G ~ orbits. The corresponding variati- 312 onal equations cutting out the transversal take a simple form and are the zeros of a map: m : E~n > j, [7] , [8] Large parts of this theory can be developed abstractly from general pro- perties of Lie groups and the fact that the curvature form of the Hermi- lian line bundle H gives the K~hler symplectic form on ~n Atiyah and Bott [I] observed that the theory of holomorphic struc- tures on a vector bundle E over an algebraic curve C could be cast in the same form, except with an infinite dimensional affine space in place of a projective space. For a holomorphic structure on E is given by a F-operator and these are parametrised by a complex affine space A . The infinite dimensional group G ~ of complex linear automorphisms of E acts by conjugation and the quotient set is by definition the set of equi- valence classes of holomorphic (or algebraic) bundles, topologically equi- valent to E Independently, and from another point of view, stability of algebraic bundles had been defined in algebraic geometry; the defini- tion uses the notion of the degree of a bundle - the integer obtained by evaluating the first Chern class on the fundamental cycle. If now E has a fixed Hermitian metric then a ~-operator induces a unique unitary connection. Regarded as connections the symmetry group of the affine space A is reduced to the subgroup GcG ~ of unitary automorphisms, and this subgroup preserves the natural metric form on the space of connections A derived from integration over C . We would have all the ingredients for the abstract theory described above if we had a Hermitian line bundle i over A with curvature generating this metric form, and acted on by G ~ It was explained above that over a space of connections we obtain vir- tual bundles from the associated elliptic operators. In particular we can take the Dirac operator over the algebraic curve C , which is the same as the ~-operator after tensoring with a square root ~I/2 of the ca- ~C nonical bundle, so the kernel and cokernel form the usual sheaf cohomolo- gy. Moreover we get a genuine line bundle if we take the highest exterior power or determinant of the relevant vector spaces. Thus we get a complex line bundle iC over A : 313 ic = X(E®K I/2) = det H0(E®K I/2 ) ® det H I (E®K I/2 ) -I C C C acted upon by G ~ , and realising via the first Chern class the cohomo- logy class obtained under our map ~ from the fundamental cycle of the curve C , as in Section I. Now Quillen has defined Hermitian metrics [9] on such determinant line bundles and computed the associated curvature to be precisely the metric form above. Thus all the ingredients for applying the general the- ory are present - the map m cutting out a transversal to the stable orbits is given by the curvature of a connection and the preferred points, minimising Quillens analytic torsion norm, are given by the projectively flat unitary connections. We can study algebraic bundles over any projective variety; in par- ticular over an algebraic surface X . Now the definition of stability requires the choice of a polarisation - the first chern class of an am- ple line bundle L over X .
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